CHAPTER 12
Extending Surface Area and Volume
0
1
Learning Targets
• Students will be able to draw isometric views of three-dimensional figures.
• Students will be able to investigate cross-sections of three-dimensional figures.
Section 12.1 Notes: Representations of Three-dimensional Figures
Vocabulary, Example
Types
Definitions, Pictures and Examples
Cross Section
Types of Slice
From the above figures,
Identify the type of shape
resulting from a vertical,
angled and horizontal
cross section of a cylinder.
Example 1:
Example 2:
Example 3:
2
Describe the shape resulting from a vertical, angled, and horizonal cross section of a square pyramid.
You Try!
Vertical:
Angled:
Horizontal:
Examples of cross
sections.
Example 4: BAKERY A customer ordered a two-layer sheet cake.
Determine the shape of each cross section of the cake below.
Example 5 :Determine the shape of the cross section shown.
a.
b.
Describe the shape resulting from each cross section.
You Try!
a.
3
b.
c.
Summary
Describe the cross section.
4
5
Geometry
Section 12.1 Worksheet
Describe each cross section.
Name:________________________________
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. A standard basketball is shaped like a sphere.
a. Draw a basketball with a vertical, angled, and horizontal slice.
b. Describe the cross section made by each slice.
6
7
Learning Targets
• Students will be able to find areas of regular polygons.
• Find areas of composite figures.
Section 11.4 Notes: Areas of Regular Polygons and Composite Figures
Vocabulary, Example
Types
Definitions, Pictures and Examples
Circumscribed
Inscribed
Regular Polygon
Apothem
Central Angle
Example 1: In the figure, pentagon PQRST is inscribed in circle X. Identify the center, a radius, an
apothem, and a central angle of the polygon. Then find the measure of a central angle.
Identify the center, radius,
apothem and central angle.
8
Example 2: The top of the table shown is a regular hexagon with a side length of 3 feet and an apothem
of 1.7 feet. What is the area of the tabletop to the nearest tenth?
Area of a Regular polygon
Example 3: The perimeter of regular polygon is 24 and apothem is 3. Find the polygon’s area.
Find the area of the regular
polygons.
Example 4: If a regular octagon with 3cm sides has an area of 60 square cm, what is the length of the
apothem?
Example 5: Find the area of the regular pentagon.
Number of Sides:_____
Central Angle:_____
Side Length:_____
Radius:_____
Apothem:______
Perimeter:______
Area:_____
9
Example 6: Find the area of the regular hexagon. Round to the nearest tenth.
Number of Sides:_____
Central Angle:_____
Side Length:_____
Radius:_____
Apothem:______
Perimeter:______
Area:_____
Example 7: Find the area of the regular pentagon. Round to the nearest tenth.
Number of Sides:_____
Central Angle:_____
Side Length:_____
Radius:_____
Apothem:______
Perimeter:______
Area:_____
You
Try!
1. Find the area of the regular triangle. Round to the nearest tenth.
Number of Sides:_____
Central Angle:_____
Side Length:_____
Radius:_____
Apothem:______
Perimeter:______
Area:_____
10
2. Find the area of the regular triangle. Round to the nearest tenth.
Number of Sides:_____
Central Angle:_____
Side Length:_____
Radius:_____
Apothem:______
Perimeter:______
Area:_____
Summary
Composite Figure
11
Example 8: The dimensions of an irregularly shaped pool are shown. What is the area of the surface of
the pool?
Example 9: Find the area of the figure in square feet.
Round to the nearest tenth if necessary
Example 10: Find the area of the shaded figure.
Example 11: Cara wants to wallpaper one wall of her family room. She has a fireplace in the center of
the wall. Find the area of the wall around the fireplace.
12
1. Detective LeRue must investigate a crime committed at the local park. How many square meters of
ground must he cover when looking for clues?
You Try!
2. Find the area and perimeter.
3. Find the area and perimeter.
JoAnn wants to lay 12” x 12” tile on her bathroom floor.
a. Find the area of the bathroom floor in her apartment floor plan.
b. If the tile comes in boxes of 15 and JoAnn buys no extra tiles,
how many boxes will she need?
Summary!
13
Geometry
Section 11.4 Regular Polygon Area Worksheet
Name:________________________________
Find the area of each regular polygon. Round to the nearest tenth.
1.
2.
Number of Sides:_____
Number of Sides:_____
Central Angle:_____
Central Angle:_____
Side Length:_____
Side Length:_____
Radius:_____
Radius:_____
Apothem:______
Apothem:______
Perimeter:______
Perimeter:______
Area:_____
Area:_____
3.
5.
4.
Number of Sides:_____
Number of Sides:_____
Central Angle:_____
Central Angle:_____
Side Length:_____
Side Length:_____
Radius:_____
Radius:_____
Apothem:______
Apothem:______
Perimeter:______
Perimeter:______
Area:_____
Area:_____
Number of Sides:_____
6.
Number of Sides:_____
Central Angle:_____
Central Angle:_____
Side Length:_____
Side Length:_____
Radius:_____
Radius:_____
Apothem:______
Apothem:______
Perimeter:______
Perimeter:______
Area:_____
Area:_____
14
15
Geometry
Section 11.4 Composite Figure Area Worksheet
Find the area of each figure. Round to the nearest tenth if necessary.
1.
2.
3.
4.
Name:________________________________
5. MINIATURE GOLF The plan for a miniature golf hole is shown below. The right angle in the drawing is a central angle. What
is the area of the playing surface? Round your answer to the nearest hundredth of a square meter.
6. TRACK A running track has an inner and outer edge. Both the inner and outer edges consist of two semicircles joined by two
straight line segments. The straight line segments are 100 yards long. The radii of the inner edge semicircles are 25 yards each and
the radii of the outer edge semicircles are 32 yards each. What is the area of the track? Round your answer to the nearest hundredth
of a yard.
7. SEMICIRCLES Bridget arranged three semicircles in the pattern shown. The right triangle has side lengths 6, 8, and 10 inches.
a. What is the total area of the three semicircles? Round your answer to the nearest hundredth
of a square inch.
b. If the right triangle had side lengths √21, √79, and 10 inches, what would the total area of the three semicircles be? Round your
answer to the nearest hundredth of a square inch.
16
17
Learning Targets
• Students will be able to find lateral areas and surface areas of prisms.
• Students will be able to find lateral areas and surface areas of cylinders.
Section 12.2 Notes: Surface Areas of Prisms and Cylinders
Vocabulary, Example
Types
Definitions, Pictures and Examples
Lateral Faces
Lateral Edges
Altitude
Base Edges
Height
Label the Parts
18
Lateral Area
Lateral Area of a Prism
Example 1:
a.
b.
c.
With a highlighter,
Identify the base of the
given figures.
Example 2: The length of each side of the base of a regular octagonal prism is 6 inches, and the height
is 11 inches. Find the lateral area.
Example 3: Lateral Area of a Prism Find the lateral area of the prism. Round your answers to the nearest
hundredth.
19
Example 4: Find the lateral area of the prism.
You Try!
Surface Area of a Prism
Surface Area of Prisms
Example 5: Find the surface area of the rectangular prism
Example 6: Find the surface area of a triangular prism. Round to the nearest tenth.
Example 7: Find the surface area of the regular hexagonal prism.
20
Example 8: Find the surface area of the regular pentagonal prism.
You Try!
Lateral Area of a Right
Cylinder
Surface area of a Cylinder
Example 9: Find the lateral area and the surface area of the cylinder. Round to the nearest thousandth.
Example 10: Find the lateral area and the surface area of the cylinder. Round to the nearest tenth.
21
You Try!
Summary!
Find the lateral area and surface area of each cylinder. Round to the nearest tenth.
a.
b.
Find the lateral area and surface area of each prism.
1.
2.
22
23
Geometry
Section 12.2 Worksheet
Name________________________
Find the lateral area and surface area of each prism. Round to the nearest tenth if necessary.
1.
2.
Lateral Area__________________
Lateral Area__________________
Surface Area___________________
Surface Area__________________
3.
4.
Lateral Area__________________
Lateral Area__________________
Surface Area___________________
Surface Area__________________
5.
6.
Lateral Area__________________
Lateral Area__________________
Surface Area___________________
Surface Area__________________
24
25
Geometry
Extra Practice 12.2
Name:_____________________________
Surface Areas of Prisms and Cylinders
Find the lateral and surface area of each prism. Round to the nearest tenth if necessary.
1.
2.
3.
4.
Find the lateral area and surface area of each cylinder. Round to the nearest tenth.
5.
7.
6.
8.
26
27
Learning Targets
•
•
Students will be able to find lateral areas and surface areas of pyramids.
Students will be able to find lateral areas and surface areas of cones.
Section 12.3 Notes: Surface Areas of Pyramids and Cones
Vocabulary, Example
Types
Definitions, Pictures and Examples
Regular Pyramid
Lateral Faces
Vertex
Lateral Edge
Slant Height
Label the Regular
Pyramid with the given
information.
28
Lateral Area of a Regular
Pyramid
Example 1: Find the lateral area of the square pyramid to the nearest tenth.
Examples in finding the
lateral area of a pyramid.
Surface Area of a Regular
Pyramid
Examples of Surface area
of pyramids
Example 2: Find the surface area of the square pyramid to the nearest tenth.
Example 3: Find the surface area of the regular pyramid. Round to the nearest hundredth.
29
You Try!
1. Find the surface area of the square pyramid.
2. Find the surface area of the regular pyramid.
Lateral Area of a Cone
Example 4: The cone represents a conical slate roof on a house. Find the lateral area.
Find the lateral area of a
cone
Example 5: Find the surface area of the cone. Round to the nearest tenth.
Find the surface area of
the cone.
30
You Try!
3. Find the surface area of the cone.
Using the solid and model provided, write in the formulas for each solid.
Solid
Model
Lateral Area
Surface Area
Prism
Cylinder
Pyramid
Cone
31
Geometry
12.3 Worksheet
Name:________________________
Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.
1.
2.
3.
4.
Find the lateral area and surface area of each cone. Round to the nearest tenth if necessary.
5.
6.
7. Find the surface area of a cone if the height is 14 centimeters and the slant height is 16.4 centimeters.
8. Find the surface area of a cone if the height is 12 inches and the diameter is 27 inches.
32
9. GAZEBOS The roof of a gazebo is a regular octagonal pyramid. If the base of the pyramid has sides of 0.5 meter and the slant
height of the roof is 1.9 meters, find the area of the roof.
10. PAPERWEIGHTS Daphne uses a paperweight shaped like a pyramid with a regular hexagon for a base. The side length of the
regular hexagon is 1 inch. The altitude of the pyramid is 2 inches.
What is the lateral surface area of this pyramid? Round your answers to the nearest hundredth
33
Learning Targets
• Students will be able to find volumes of prisms.
• Students will be able to find volumes of cylinders.
Section 12.4 Notes: Volumes of Prisms and Cylinders
Vocabulary, Example
Types
Definitions, Pictures and Examples
Volume of a Prism
Example 1: Find the volume of the prism. Make sure to label the correct units.
Example 2: Find the volume of the prism. Make sure you label the correct units.
You Try!
1. Find the volume of the prism. Make sure you label the correct units.
34
Volume Of a Cylinder
Example 2: Find the volume of the cylinder. Make sure to label the correct units and leave your answer
in terms of π.
You Try!
Find the volume of the cylinder.
Example 3: Find the volume of the oblique cylinder.
Label your units and leave your answer in terms of π.
35
Find the volume of the given prism.
Summary!
Find the volume of the cylinder.
36
37
Geometry
12.4 Worksheet
Name:_____________________________
Find the volume of each prism or cylinder. Round to the nearest tenth if necessary.
1. V=__________
2. V=__________
3. V=__________
4. V=__________
5. V=__________
6. V=__________
1
7. AQUARIUM Mr. Gutierrez purchased a cylindrical aquarium for his office. The aquarium has a height of 25 inches and a radius
2
of 21 inches.
a. What is the volume of the aquarium in cubic feet?
b. If there are 7.48 gallons in a cubic foot, how many gallons of water does the aquarium hold?
c. If a cubic foot of water weighs about 62.4 pounds, what is the weight of the water in the aquarium to the "nearest five pounds?
38
39
Learning Targets
• Students will be able to find volumes of pyramids.
• Students will be able to find volumes of cones
Section 12.5 Notes: Volumes of Pyramids and Cones
Vocabulary, Example
Types
Definitions, Pictures and Examples
Volume of a Pyramid
Example 1: Find the volume of the square pyramid. Label the units.
Example 2: Find the volume of the triangular pyramid. Label the units.
1. Find the volume of the pentagonal pyramid. Label the units.
You Try!
Volume of a Cone
40
Example 3: Find the volume of the cone. Leave your answers in terms of π and label the units.
Example 4: Find the volume of the oblique cone. Round your answers to the tenth.
1. Find the volume of each cone.
You Try!
In the below chart, fill out the volume formulas for each individual solid.
Solid
Prism
Cylinder
Pyramid
Cone
Model
Volume
41
Geometry
12.5 Worksheet
Name:____________________________
Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.
1. V=__________
2. V=__________
3. V=__________
4. V=__________
5. V=__________
6. V=__________
7. CONSTRUCTION Mr. Ganty built a conical storage shed. The base of the shed is 4 meters in diameter and the height of the shed
is 3.8 meters. What is the volume of the shed?
8. HISTORY The start of the pyramid age began with King Zoser’s pyramid, erected in the 27th century B.C. In its original state, it
stood 62 meters high with a rectangular base that measured 140 meters by 118 meters. Find the volume of the original pyramid.
9. SCULPTING A sculptor wants to remove stone from a cylindrical block 3 feet high and turn it into a cone. The diameter of the
base of the cone and cylinder is
2 feet.
What is the volume of the stone that the sculptor must remove? Round your answer to the nearest hundredth.
42
10. STAGES A stage has the form of a square pyramid with the top sliced off along a plane parallel to the base. The side length of the
top square is 12 feet
and the side length of the bottom square is 16 feet.
The height of the stage is 3 feet.
a. What is the volume of the entire square pyramid that the stage is part of?
b. What is the volume of the top of the pyramid
that is removed to get the stage?
c. What is the volume of the stage?
43
Learning Targets
• Students will be able to find surface areas of spheres.
• Students will be able to find volumes of spheres.
Section 12.6 Notes: Surface Areas and Volumes of Spheres
Vocabulary, Example
Types
Definitions, Pictures and Examples
Surface Area of a Sphere
Example 1: Find the surface area of the sphere. Round to the nearest tenth and label the units.
Find the surface area
A plane can intersect a sphere in a point or in a circle. If the circle contains the center of the sphere, the
intersection is called a _______________________. The endpoints of a diameter of a great circle are
called the _________.
Since a great circle has the same center as the sphere and its radii are also radii of the sphere, it is the
largest circle that can be drawn on a sphere. A great circle separates a sphere into two congruent halves,
called ________________.
Surface Area of a
Hemisphere
Example 2: Find the surface area of the hemisphere. Label the units.
44
Example 3: Find the surface area of the sphere if the circumference of the great circle is 10π.
Example 4: Find the surface area of the sphere if the area of the great circle is approximately 160 square
meters.
Volume of a Sphere
Example 5: Find the volume of the sphere. Round to the nearest hundredth and label the units.
Example 6: Find the volume of the sphere with a great circle that has a circumference of 30π
centimeters. Round to the nearest tenth.
45
Volume of a Hemisphere
Example 7: Find the volume of the hemisphere with a diameter of 6 ft.
Example 8 RECESS The jungle gym outside of Jada’s school is a perfect hemisphere. It has a volume
of 4,000π cubic feet. What is the diameter of the jungle gym?
46
1
Geometry
Name:________________________________
12.6 Worksheet
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
1. SA=______________
2. SA=______________
3. hemisphere: radius of great circle = 8.4 in.
4. sphere: area of great circle ≈ 29.8 m2
Find the volume of each sphere or hemisphere. Round to the nearest tenth.
5. V=______________
6. V=_____________
7. hemisphere: diameter = 18 mm
7. V=_____________
8. sphere: circumference ≈ 36 yds
8. V=_____________
9. sphere: radius = 12.4 in.
9. V=_____________
2
10. CUBES Marcus builds a sphere inside of a cube.
The sphere fits snugly inside the cube so that the sphere touches the cube at one point on each side.
The side length of the cube is 2 inches.
a. What is the surface area of the cube?
b. What is the surface area of the sphere?
Round your answers to the nearest hundredth.
c. What is the ratio of the surface area of the cube to the surface area of the sphere? Round your answer to the nearest hundredth
3
Chapter 12 Extension Surface Area of Composite Figures
Vocabulary, Example
Types
Definitions, Pictures and Examples
Surface Area of a
Composite Figure
Identify the solids that
form the composite solid.
1.
2.
3.
Example 1: Identify the solids that make the composite solid and then find the surface area.
4
Example 2: Identify the solids that make the composite solid and then find the surface area.
Example 3: Steven is making a wood box to fit two flower pots in the middle. His rough draft of the
contraption looks like the diagram below. Find the amount of paint he needs to paint the outside and the
inside holes.
Example 4: Identify the solids that make the composite figure. Find the total surface area and volume of
the composite space figures.
5
Example 5: Identify the solids that make the composite figure. Find the volume of the composite figure.
1. Find the total surface area and volume of the composite figure.
You Try!
2. A storage bin is shaped as in the figure. The radius of the cylindrical top is 7 ft. The overall height of
the bin is 26 ft. and the altitude of the conical section is 12 ft. Find the total surface area of the bin in
bushels.
6
3. The dimensions of the base of a rectangular solid are 5 and 8, and its altitude is 12. A
hole, extending from upper base to lower base, is in the shape of a right triangular prism
whose bases are equilateral ∆’s having an edge of 3. Determine the total surface area of
the figure.
4. Jimmy’s lunch box in the shape of a half cylinder on a rectangular box.
Find the total area of metal needed to manufacture it
8
6
10
5. A rectangular prism is 40 ft by 38 ft by 15 ft. Shown below
is the prism with a half cylinder removed. Find the volume of the original prism remaining.
15 ft
38 ft
40 ft
12 ft
10 ft
10 ft
7